Interpreting a period-adding bifurcation scenario in neural bursting patterns using border-collision bifurcation in a discontinuous map of a slow control variable

To further identify the dynamics of the period-adding bifurcation scenarios observed in both biological experiment and simulations with the differential Chay model, this paper fits a discontinuous map of a slow control variable of the Chay model based on simulation results. The procedure of period adding bifurcation scenario from period k to period k + 1 bursting (k = 1, 2, 3, 4) involved in the period-adding cascades and the stochastic effect of noise near each bifurcation point is also reproduced in the discontinuous map. Moreover, dynamics of the border-collision bifurcation are identified in the discontinuous map, which is employed to understand the experimentally observed period increment sequence. The simple discontinuous map is of practical importance in the modeling of collective behaviours of neural populations like synchronization in large neural circuits.

[1]  Christiansen,et al.  Phase diagram of a modulated relaxation oscillator with a finite resetting time. , 1992, Physical review. A, Atomic, molecular, and optical physics.

[2]  K. Schäfer,et al.  Oscillation and noise determine signal transduction in shark multimodal sensory cells , 1994, Nature.

[3]  Michael Schanz,et al.  Codimension-three bifurcations: explanation of the complex one-, two-, and three-dimensional bifurcation structures in nonsmooth maps. , 2007, Physical review. E, Statistical, nonlinear, and soft matter physics.

[4]  Da-Ren He,et al.  Intermittency between type I and type v in some quasi-discontinuous systems , 2001 .

[5]  Paul C. Bressloff,et al.  Neuronal dynamics based on discontinuous circle maps , 1990 .

[6]  Xu Jian-Xue,et al.  Propagation of periodic and chaotic action potential trains along nerve fibers , 1997 .

[7]  J. Hindmarsh,et al.  A model of neuronal bursting using three coupled first order differential equations , 1984, Proceedings of the Royal Society of London. Series B. Biological Sciences.

[8]  Eugene M. Izhikevich,et al.  Neural excitability, Spiking and bursting , 2000, Int. J. Bifurc. Chaos.

[9]  Mannella,et al.  Fast and precise algorithm for computer simulation of stochastic differential equations. , 1989, Physical review. A, General physics.

[10]  Georgi S Medvedev,et al.  Transition to bursting via deterministic chaos. , 2006, Physical review letters.

[11]  Gábor Stépán,et al.  Dynamics of Piecewise Linear Discontinuous Maps , 2004, Int. J. Bifurc. Chaos.

[12]  Teresa Ree Chay,et al.  Chaos in a three-variable model of an excitable cell , 1985 .

[13]  HE Da-Ren,et al.  Characteristics of Period-Doubling Bifurcation Cascades in Quasi-discontinuous Systems ∗ , 2001 .

[14]  A. Sharkovsky,et al.  Chaos in Some 1-D Discontinuous Maps that Apper in the Analysis of Electrical Circuits , 1993 .

[15]  Pawel Hitczenko,et al.  Bursting Oscillations Induced by Small Noise , 2007, SIAM J. Appl. Math..

[16]  Qishao Lu,et al.  Two-parameter bifurcation analysis of firing activities in the Chay neuronal model , 2008, Neurocomputing.

[17]  Ott,et al.  Border-collision bifurcations: An explanation for observed bifurcation phenomena. , 1994, Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics.

[18]  D. Terman,et al.  The transition from bursting to continuous spiking in excitable membrane models , 1992 .

[19]  S. Coombes,et al.  Period-adding bifurcations and chaos in a periodically stimulated excitable neural relaxation oscillator. , 2000, Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics.

[20]  Qishao Lu,et al.  Integer multiple spiking in neuronal pacemakers without external periodic stimulation , 2001 .

[21]  Mario di Bernardo,et al.  C-bifurcations and period-adding in one-dimensional piecewise-smooth maps , 2003 .

[22]  A. Kawczynski,et al.  Period-Adding Bifurcations in Mixed-Mode Oscillations in the Belousov-Zhabotinsky Reaction at Various Residence Times in a CSTR , 2001 .

[23]  Ni Fei,et al.  Stochastic period-doubling bifurcation analysis of a Rössler system with a bounded random parameter , 2010 .

[24]  Arun V. Holden,et al.  Bifurcations, burstings, chaos and crises in the Rose-Hindmarsh model for neuronal activity , 1993 .

[25]  D. He,et al.  Multiple devil's staircase and type-V intermittency , 1998 .

[26]  James A. Yorke,et al.  Border-collision bifurcations including “period two to period three” for piecewise smooth systems , 1992 .

[27]  Georgi S. Medvedev,et al.  Reduction of a model of an excitable cell to a one-dimensional map , 2005 .

[28]  A Garfinkel,et al.  Evidence for a novel bursting mechanism in rodent trigeminal neurons. , 1998, Biophysical journal.

[29]  H. Gu,et al.  Qualitatively different bifurcation scenarios observed in the firing of identical nerve fibers , 2009 .

[30]  Stephen John Hogan,et al.  Dynamics of a piecewise linear map with a gap , 2006, Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences.

[31]  V. N. Belykh,et al.  Homoclinic bifurcations leading to the emergence of bursting oscillations in cell models , 2000 .

[32]  J. Rinzel,et al.  Dissection of a model for neuronal parabolic bursting , 1987, Journal of mathematical biology.

[33]  Andrey Shilnikov,et al.  Transition between tonic spiking and bursting in a neuron model via the blue-sky catastrophe. , 2005, Physical review letters.

[34]  Teresa Ree Chay,et al.  Generation of periodic and chaotic bursting in an excitable cell model , 1994, Biological Cybernetics.

[35]  Soumitro Banerjee,et al.  On the existence of low-period orbits in n-dimensional piecewise linear discontinuous maps , 2008 .

[36]  Arun V. Holden,et al.  Crisis-induced chaos in the Rose-Hindmarsh model for neuronal activity , 1992 .

[37]  Li Li,et al.  Dynamics of autonomous stochastic resonance in neural period adding bifurcation scenarios , 2003 .

[38]  Michael Schanz,et al.  Border-collision period-doubling scenario. , 2004, Physical review. E, Statistical, nonlinear, and soft matter physics.

[39]  Li Li,et al.  Identifying Distinct Stochastic Dynamics from Chaos: a Study on Multimodal Neural Firing Patterns , 2009, Int. J. Bifurc. Chaos.

[40]  James A. Yorke,et al.  BORDER-COLLISION BIFURCATIONS FOR PIECEWISE SMOOTH ONE-DIMENSIONAL MAPS , 1995 .

[41]  Frank Moss,et al.  Noise enhancement of information transfer in crayfish mechanoreceptors by stochastic resonance , 1993, Nature.

[42]  Michael Schanz,et al.  Period-Doubling Scenario without flip bifurcations in a One-Dimensional Map , 2005, Int. J. Bifurc. Chaos.

[43]  Da-Ren He,et al.  A multiple devil's staircase in a discontinuous map , 1997 .

[44]  Teresa Ree Chay,et al.  BURSTING, SPIKING, CHAOS, FRACTALS, AND UNIVERSALITY IN BIOLOGICAL RHYTHMS , 1995 .

[45]  Soumitro Banerjee,et al.  Border-Collision bifurcations in One-Dimensional Discontinuous Maps , 2003, Int. J. Bifurc. Chaos.

[46]  San-Jue Hu,et al.  PERIOD-ADDING BIFURCATION WITH CHAOS IN THE INTERSPIKE INTERVALS GENERATED BY AN EXPERIMENTAL NEURAL PACEMAKER , 1997 .

[47]  Andrey Shilnikov,et al.  Origin of bursting through homoclinic spike adding in a neuron model. , 2007, Physical review letters.

[48]  Ren Wei,et al.  Two Different Bifurcation Scenarios in Neural Firing Rhythms Discovered in Biological Experiments by Adjusting Two Parameters , 2008 .

[49]  Li Li,et al.  A Series of bifurcation Scenarios in the Firing Pattern Transitions in an Experimental Neural pacemaker , 2004, Int. J. Bifurc. Chaos.